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A random variable is a fundamental concept in probability theory and statistics. It is essentially a variable whose possible values are numerical outcomes of a random phenomenon. In the context of the exercise, the random variable \(X\) represents the yearly revenue that could be gained from a product based on its level of success. Each outcome of \(X\) corresponds to a different revenue amount, given as:

• Very successful: \(10\) million dollars
• Moderately successful: \(5\) million dollars
• Unsuccessful: \(1\) million dollars

These different outcomes are associated with specific probabilities, which helps in predicting the likelihood of each revenue instance occurring. Understanding random variables is crucial as they help us model and analyze real-world processes that involve uncertainty.

A probability distribution provides a way to model and describe the underlying random process of a random variable. It assigns probabilities to each of the potential values the random variable can take. For discrete random variables, like in our exercise, we often use a Probability Mass Function (PMF).

In the case of the random variable \(X\), which represents the product's yearly revenue, the PMF assigns probabilities to each outcome:

• \(P(X = 10) = 0.3\)
• \(P(X = 5) = 0.6\)
• \(P(X = 1) = 0.1\)

The sum of all the probabilities in a PMF must equal 1. This assures us that one of the outcomes will occur. Studying probability distributions offers valuable insights into how likely different scenarios are, which aids in strategic planning and decision-making.

The expected value is a key concept used to summarize a probability distribution. It provides a measure of the 'center' of the distribution, often seen as the 'long-term average' outcome if a random process is repeated many times.

For a discrete random variable like our revenue example, the expected value \(E(X)\) is calculated by multiplying each possible outcome by its probability and summing the results: \[E(X) = (10 \times 0.3) + (5 \times 0.6) + (1 \times 0.1)\]\[E(X) = 3 + 3 + 0.1 = 6.1\]Therefore, the expected yearly revenue of the product is \(6.1\) million dollars. The expected value offers an estimate of the potential returns, acting as a guide for making decisions where outcomes are uncertain.